Consider the infinite geometric sequence, defined as
$$a_n=a_0|r|^{|n|} : r \in \mathbb{Z}^+$$
As the number of terms in the sequence is unbounded, it seems reasonable that one can define a term in the sequence that is indexed by a transfinite ordinal. Is there any set theoretical objection to defining terms such as $a_{\omega_0}, a_{\omega_1}, a_{\omega_2}, \dots$, therefore implying values such as $a_{\omega_0}=a_0|r|^{\aleph_0}$? This question is particularly motivated by a binary tree and the consideration of whether or not an $\omega_0$th row exists within the tree.
I had a bit of a debate with the editor of a journal (an expert in set theory) recently about this. He was reluctant to permit such a definition but was evasive in stating his reasoning.
Short version: sort of but not very satisfyingly.
For $r,a,n$ possibly infinite cardinals, cardinal multiplication and cardinal exponentiation do indeed give meaning to the expression $r\cdot a^n$. However, fixing $r$ and $a$, as a function of $n$ this isn't very interesting since cardinal arithmetic is so "coarse." For example, as long as $a,r\le 2^{\aleph_0}$ (and $r>1$) we'll always have $a\cdot r^{\aleph_0}=2^{\aleph_0}$ since $$2^{\aleph_0}\cdot (2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}\cdot 2^{\aleph_0\cdot \aleph_0}=2^{\aleph_0}.$$ So things get pretty boring.
Ordinal arithmetic is somewhat more interesting than cardinal arithmetic, but it still trivializes a lot in this case. In particular, for any finite $n$ we have $$n^\omega=\sup_{k\in\omega}n^k=\omega$$ in the sense of ordinal exponentiation.
This is an instance of a more general situation: "ordinary" mathematical operations often only extend into the transfinite at the cost of losing lots of the original nature. Cardinal exponentiation is a good positive example of this: it is incredibly interesting, but completely fails to look like ordinary exponentiation - and indeed the key ingredient which makes cardinal exponentiation interesting (namely cofinality) doesn't even make sense at the finite level.